×

Scrolls and hyperbolicity. (English) Zbl 1270.14027

Several questions concerning very general hypersurfaces of degree \(d\) in the projective space \(\mathbb{P}^n\) are of interest, for instance the following are discussed in the paper under review: (1) to know what is the lowest geometric genus of a reduced and irreducibe curve lying on these hypersurfaces; (2) to bound the geometric genus (or other numerical invariants) of higher-dimensional subvarieties of them; (3) to know if they do not admit a non constant morphism from an abelian variety, algebraic hyperbolicity; (4) to know if they do not admit a non-constant entire curve, Kobayashi hyperbolicity; (5) to know if there exists a real number \(\epsilon>0\) such that any algebraic curve on the hypersurface verifies \(2g(C)-2\geq \epsilon \deg C\), Demailly algebraic hyperbolicity.
In this paper, the authors use a degeneration method of the general hypersurface to a special one (mainly scrolls) to deal with this kind of questions. In particular they provide a proof of the non-existence of low genera curves on general surfaces of degree \(\geq 5\) of the three-dimensional projective space (see Section 2). They also provide a bound of the geometric genera of surfaces on general threefolds in \(\mathbb{P}^4\) (see Section 3). Finally, they prove the existence of hyperbolic surfaces (resp. threefolds) of any degree \(d \geq 6\) (resp. \(d\geq 12\)), being unknown the case \(d=7\), in \(\mathbb{P}^3\) (resp. \(\mathbb{P}^4\)), showing consequently their Demailly algebraic hyperbolictiy (see Theorem 4.6 in Section 4).

MSC:

14N25 Varieties of low degree
14J70 Hypersurfaces and algebraic geometry
32J25 Transcendental methods of algebraic geometry (complex-analytic aspects)
32Q45 Hyperbolic and Kobayashi hyperbolic manifolds
PDFBibTeX XMLCite
Full Text: DOI arXiv

References:

[1] DOI: 10.1007/BF03016613 · JFM 54.0685.02
[2] Arbarello E., Rend. Sem. Mat. Univ. Politec. Torino 38 pp 87–
[3] DOI: 10.1007/BF01679702 · Zbl 0454.14023
[4] DOI: 10.1007/BFb0085919
[5] DOI: 10.4171/064
[6] Bertin M.-A., Le Matematiche 53 pp 15–
[7] Bogomolov F., Soviet Math. Dokl. 18 pp 1294–
[8] Bonnesen P., Bull. Acad. Royal Sci. Lett. Danemarque 4 pp 281–
[9] Brody R., Trans. Amer. Math. Soc. 235 pp 213–
[10] Calabri A., Rend. Lincei Mat. Appl. 17 pp 95–
[11] Chen X., J. Algebraic Geom. 8 pp 245–
[12] DOI: 10.1017/S0305004100004904 · Zbl 1068.14508
[13] DOI: 10.1090/S0002-9947-2011-05401-2 · Zbl 1227.14022
[14] Clemens H., Ann. Sci. École Norm. Sup. (4) 19 pp 629– · Zbl 0611.14024
[15] DOI: 10.1353/ajm.2000.0019 · Zbl 0966.32014
[16] DOI: 10.1007/s00222-010-0232-4 · Zbl 1192.32014
[17] Diverio S., J. Reine Angew. Math. 649 pp 55–
[18] DOI: 10.1017/CBO9781139084437 · Zbl 1252.14001
[19] DOI: 10.1007/s00208-004-0551-0 · Zbl 1071.14045
[20] DOI: 10.1007/BF01394349 · Zbl 0701.14002
[21] DOI: 10.1007/BF01446583 · Zbl 0746.14019
[22] Enriques F., Le Superficie Algebriche (1949)
[23] DOI: 10.1007/978-1-4612-1700-8
[24] DOI: 10.1007/BF03191236 · Zbl 1230.14054
[25] DOI: 10.1007/978-1-4757-3849-0
[26] DOI: 10.1016/S0019-3577(08)80001-8 · Zbl 1168.14021
[27] DOI: 10.1007/978-3-662-03662-4
[28] DOI: 10.4310/MRL.1995.v2.n6.a1 · Zbl 0870.14020
[29] DOI: 10.1007/s000390050091 · Zbl 0951.14014
[30] Mezzetti E., An. Ştiinţ. Univ. Ovidius Constanţa Ser. Mat. 5 pp 51–
[31] Moishezon B., Lecture Notes in Mathematics 630, in: Complex Surfaces and Connected Sums of Complex Projective Planes (1977) · Zbl 0392.32015
[32] D. R. Morrison, Topics in Transcendental Algebraic Geometry, Annals of Mathematics Studies 106 (Princeton University Press, 1984) pp. 101–119.
[33] Nobile A., Ann. Sci. École Norm. Sup. (4) 20 pp 465– · Zbl 0687.14024
[34] DOI: 10.1090/S1056-3911-02-00328-4 · Zbl 1054.14057
[35] DOI: 10.1090/S0002-9947-03-03250-1 · Zbl 1056.14060
[36] DOI: 10.1016/j.matpur.2010.10.008 · Zbl 1235.32014
[37] DOI: 10.1007/s00208-007-0172-5 · Zbl 1137.32010
[38] DOI: 10.1007/BFb0062933 · Zbl 0391.14008
[39] DOI: 10.1007/BF03017734 · JFM 32.0648.04
[40] Shiffman B., Internat. J. Math. 11 pp 65–
[41] Shiffman B., Houston J. Math. 28 pp 377–
[42] DOI: 10.1007/s10688-005-0020-x · Zbl 1095.32009
[43] Shin D., Osaka J. Math. 44 pp 1–
[44] DOI: 10.1353/ajm.1997.0033 · Zbl 0947.32012
[45] Voisin C., J. Differential Geom. 44 pp 200– · Zbl 0883.14022
[46] Voisin C., J. Differential Geom. 49 pp 601– · Zbl 0994.14026
[47] DOI: 10.4310/AJM.2000.v4.n2.a7 · Zbl 0972.14032
[48] Xu G., J. Differential Geom. 39 pp 139– · Zbl 0823.14030
[49] DOI: 10.1090/S0002-9947-96-01613-3 · Zbl 0871.14037
[50] DOI: 10.1070/SM1989v063n02ABEH003278 · Zbl 0668.32023
[51] DOI: 10.1007/s10688-009-0015-0 · Zbl 1271.32028
[52] Zak F. L., Translations of Mathematical Monographs 127, in: Tangents and Secants of Algebraic Varieties (1993) · Zbl 0795.14018
[53] DOI: 10.2307/2374074 · Zbl 0516.14023
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.